Is it Possible to Get a SAR Image of Ocean Waves Free of Spectral Cut-off by Direct Measurement?

A method of SAR sounding of the ocean surface is proposed, which is capable of providing an undistorted by spectral cut-off wave pattern. The method involves the use of two synchronized SARs, which look across the track line and illuminate the same area of the surface. Each SAR records its own backscattered signal, which are then multiplied with each other and the resulting signal is undergoing the procedure of matched filtering. Numerical estimates of the applicability conditions of the method are carried out.

On the behavior of 1-Laplacian ratio cuts on nearly rectangular domains

The $p$-Laplacian has attracted more and more attention in data analysis disciplines in the past decade. However, there is still a knowledge gap about its behavior, which limits its practical application. In this paper, we are interested in its iterative behavior in domains contained in two-dimensional Euclidean space. Given a connected set $\varOmega _0 \subset \mathbb{R}^2$, define a sequence of sets $(\varOmega _n)_{n=0}^{\infty }$ where $\varOmega _{n+1}$ is the subset of $\varOmega _n$ where the first eigenfunction of the (properly normalized) Neumann $p$-Laplacian $ -\varDelta ^{(p)} \phi = \lambda _1 |\phi |^{p-2} \phi $ is positive (or negative). For $p=1$, this is also referred to as the ratio cut of the domain. We conjecture that these sets converge to the set of rectangles with eccentricity bounded by 2 in the Gromov–Hausdorff distance as long as they have a certain distance to the boundary $\partial \varOmega _0$. We establish some aspects of this conjecture for $p=1$ where we prove that (1) the 1-Laplacian spectral cut of domains sufficiently close to rectangles is a circular arc that is closer to flat than the original domain (leading eventually to quadrilaterals) and (2) quadrilaterals close to a rectangle of aspect ratio $2$ stay close to quadrilaterals and move closer to rectangles in a suitable metric. We also discuss some numerical aspects and pose many open questions.

The December 2015 re-brightening of V404 Cygni

In December 2015 the black hole binary V404 Cyg underwent a secondary outburst after the main June 2015 event. We monitored this re-brightening with the INTEGRAL and Swift satellites, and in this paper we report the results of the time-resolved spectral analysis of these data. The December outburst shared several characteristics with the June event. The well-sampled INTEGRAL light curve shows up to ten Crab flares, which are separated by relatively weak non-flaring emission phases when compared to the June outburst. The spectra are nicely described by absorbed Comptonization models, with hard photon indices, Γ ≲ 2, and significant detections of a high-energy cut-off only during the bright flares. This is in contrast to the June outburst, where the Comptonization models gave electron temperatures mostly in the 30–50 keV range, while some spectra were soft (Γ ~ 2.5) without signs of any spectral cut-off. Similarly to the June outburst, we see clear signs of a variable local absorber in the soft energy band covered by Swift/XRT and INTEGRAL/JEM-X, which causes rapid spectral variations observed during the flares. During one flare, both Swift and INTEGRAL captured V404 Cyg in a state where the absorber was nearly Compton thick, N H ≈ 1024 cm−2, and the broad-band spectrum was similar to obscured AGN spectra, as seen during the X-ray plateaus in the June outburst. We conclude that the spectral behaviour of V404 Cyg during the December outburst was analogous with the first few days of the June outburst, both having hard X-ray flares that were intermittently influenced by obscuration due to nearly Compton-thick outflows launched from the accretion disc.

Hard Spectral Tails in Magnetars

Pulsed non-thermal quiescent emission between 10 keV and around 150 keV has been observed in ~10 magnetars. For inner magnetospheric models of such hard X-ray signals, resonant Compton upscattering of soft thermal photons from the neutron star surface is the most efficient radiative process. We present angle-dependent hard X-ray upscattering model spectra for uncooled monoenergetic relativistic electrons. The spectral cut-off energies are critically dependent on the observer viewing angles and electron Lorentz factor. We find that electrons with energies less than around 15 MeV will emit most of their radiation below 250 keV, consistent with the observed turnovers in magnetar hard X-ray tails. Moreover, electrons of higher energy still emit most of the radiation below around 1 MeV, except for quasi-equatorial emission locales for select pulses phases. Our spectral computations use new state-of-the-art, spin-dependent formalism for the QED Compton scattering cross section in strong magnetic fields.

Logarithms and sectorial projections for elliptic boundary problems

On a compact manifold with boundary, consider the realization $B$ of an elliptic, possibly pseudodifferential, boundary value problem having a spectral cut (a ray free of eigenvalues), say $\mathsf{R}_{-}$. In the first part of the paper we define and discuss in detail the operator $\log B$; its residue (generalizing the Wodzicki residue) is essentially proportional to the zeta function value at zero, $\zeta (B,0)$, and it enters in an important way in studies of composed zeta functions $\zeta (A,B,s)= {\operatorname {Tr}}(AB^{-s})$ (pursued elsewhere). There is a similar definition of the operator $\log_{\theta}B$, when the spectral cut is at a general angle $\theta$. When $B$ has spectral cuts at two angles $\theta <\varphi$, one can define the sectorial projection $\Pi_{\theta,\varphi} (B)$ whose range contains the generalized eigenspaces for eigenvalues with argument in $\left]\theta,\varphi \right[$; this is studied in the last part of the paper. The operator $\Pi_{\theta ,\varphi}(B)$ is shown to be proportional to the difference between $\log_{\theta}B$ and $\log_{\varphi} B$, having slightly better symbol properties than they have. We show by examples that it belongs to the Boutet de Monvel calculus in many special cases, but lies outside the calculus in general.